\(\alpha <0\)  refers to the occupied bands (eigenenstates of \(H\left(k_{y},k\right) \)  below the Fermi level). \(Q^{-1}\left( k\right) =Q\left(k\right) \) , as \(Q^{2}=\mathit {I}_{N}\) . Winding number describes the
topological properties near the zero energy (Fermi level) of 1D bulk
Hamiltonians \(H\left( k_{y},k\right) \)  with the chiral symmetry operator \(S\)[1]} . Here we would like to stress again one should not confuse \(k_{y}\)  with \(k\) , since \(k_{y}\)  appearing in the bulk Hamiltonian \(H\left(k_{y},k\right) \)  is a parameter of the system and \(k\)  is the wave vector of
the effective SSH chain when its length is taken to be infinite. The
integration appearing in the definition of \(W\)  is over \(k\) , with \(W\)  being a
function of \(k_{y}\) . The zero-energy edge states (flat bands) exist in the
restricted region \(k_{y}\in \left[ -\pi ,-\frac{2}{3}\pi \right] \cup \left[\frac{2}{3}\pi ,\pi \right] \)  and correspond \(W=1\) , which agree with
previous literatures [2]}, [3]}. The effective 1D Hamiltonian \(H_{\mathrm {zig}}\left( k_{y}\right) \)  for a MLG with a zigzag edge can be
obtained by switching \(w\)  and \(v\) , \(a_{n\text{ }}\) and \(b_{n}\)  in \(H_{\mathrm {bea}}\left( k_{y}\right) \)  as zigzag edge is related with bearded edge
through an exchange of basis and coupling (see Fig. REF (a) and (b)).
In terms of their corresponding SSH chain, zigzag nanoribbon and bearded
nanoribbon differ from each other by a switch between intercell coupling and
intracell coupling of the chain. It can be observed from Fig. REF (a)
and (b) that zero-energy edge states (flat bands) of \(H_{\mathrm {zig}}\left(k_{y}\right) \)  and non-zero winding number of \(H_{\mathrm {zig}}\left(k_{y},k\right) \)  appear in complementary region of \(k_{y}\)  to the one with
bearded edges.
<FIGURE>